This work introduces a novel approach for analyzing the uniform practical stability (UPS) and strong uniform practical stability (SUPS) of Caputo fractional dynamic equations on time scales, using two measures ( $m,m_0$). The method employs an extended derivative, the Caputo fractional delta Dini derivative (CFr $\mathrm{\Delta }$DiD) of order $\zeta \in (0,1)$, addressing the gap in unified stability frameworks for fractional hybrid systems that span both continuous and discrete time domains. This generalized framework not only unifies various stability concepts but also makes it applicable to hybrid systems with both gradual and abrupt changes. The UPS and SUPS results are demonstrated through illustrative examples.

This work presents new Kneser-type oscillation criteria for second-order quasilinear functional dynamic equations defined on arbitrary unbounded above time scales. Our approach employs the Riccati transformation technique in conjunction with the integral averaging method. The results show a significant improvement over recent Kneser-type oscillation criteria. We provided several illustrative examples to highlight the importance of our findings.

We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type $(p,q)$-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the $(p,q)$-fractional setting.

This study introduces the t-arbicular fuzzy (t-AF) set, an extension of the t-spherical fuzzy set, to enhance decision-making in complex environments. The research focuses on the theoretical foundation of the t-AF set, encompassing the development of algebraic operations and comparison rules. In addition, we propose novel aggregation operators (AOs), including the t-AF weighted average (t-AFWA) and t-AF weighted geometric (t-AFWG) operators. Key properties such as idempotency, monotonicity, and boundedness of these operators are thoroughly examined. Furthermore, distance measures are formulated alongside their essential characteristics and special cases. To address multi-criteria group decision-making (MCGDM) problems under t-AF environment with unknown weight information, the tomada de decisao interativa multicriterio (TODIM) method is integrated with the criteria importance through an intercriteria correlation (CRITIC) approach. Finally, the proposed methodologies are validated through a case study on selecting the optimal gate security system, demonstrating their effectiveness and applicability in real-world scenarios.

In this article, we develop the Laplace transform (LT) based Chebyshev spectral collocation method (CSCM) to approximate the time fractional advection-diffusion equation, incorporating the Atangana-Baleanu Caputo (ABC) derivative. The advection-diffusion equation, which governs the transport of mass, heat, or energy through combined advection and diffusion processes, is central to modeling physical systems with nonlocal behavior. Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain, eliminating the need for classical time-stepping methods that often suffer from stability constraints. For spatial discretization, we employ the CSCM, where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes, achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods. Finally, the solution is reverted to the time domain using contour integration technique. We also establish the existence and uniqueness of the solution for the proposed problem. The performance, efficiency, and accuracy of the proposed method are validated through various fractional advection-diffusion problems. The computed results demonstrate that the proposed method has less computational cost and is highly accurate.